The objective of this research is to develop and apply new estimation, testing and regression procedures for analyzing left-truncated and right-censored survival data in the context of two medical studies, with the following aims: to develop more efficient and more flexible biostatistical methods that will be generally useful in research on cancer as well as in other research areas: and to apply the developed methods towards analysis of effects of bone marrow transplants (BMTAX) and comparison of BMT with traditional chemotherapy as a postremission treatment in patients with leukemia, and towards a study of efficacy of polyvalent melanoma cell vaccine in post- operative melanoma patients. The proposed research will be divided into three projects according to the following topics for analysis of left- truncated and right-censored data: 1) semiparametric estimation of survival probabilities: 2) nonparametric and semiparametric two-sample tests; and 3) semiparametric and nonparametric additive risk transformation models for survival regression. Semiparametric maximum likelihood estimation will be studied in project 1 and the developed estimate is expected to be more efficient that the traditional product limit estimate. In project 2, a quantily comparison function based on the percentile-percentile plot will be used to derive nonparametric and semiparametric two-sample tests which are expected to be more powerful than logrank type tests under crossing hazards alternatives. Additive risk transformation models will be introduced and studied in project 3, which will provide useful alternatives to the Cox proportional hazards mode when the proportional hazards assumption does not hold.